Simplified energy landscape of the $\phi^4$ model and the phase transition

The on lattice $\phi^4$ model is a paradigmatic example of continuous real variables model undergoing a continuous symmetry braking phase transition (SBPT). In this paper we study the $\mathbb{Z}_2$-symmetric mean-field version without the quadratic term of the local potential. Obviously, the simplification is directly extensible to the other symmetry groups for which the model undergoes a SBPT. We show that the $\mathbb{Z}_2$-SBPT is not affected by the quadratic term, and that the potential energy landscape turns out greatly simplified. In particular, there exist only three critical points, to confront with an amount growing as $e^N$ ($N$ is the number of degrees of freedom) of the model with non-vanishing quadratic term. In our opinion, this is an crucial feature because in recent years the study of the link between statistical mechanic and geometric-topological properties of configuration space has received an increasing attention. In this paper we study the equipotential hypersurfaces with the aim of deepening our understanding of the link between SBPTs and the truly essential geometric-topological properties of the energy potential landscape.